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Manuscript/sfBSE.tex
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@ -762,6 +762,7 @@ In the last panel we have results for BSE calculation with and without spin-flip
\begin{figure*} \begin{figure*}
\includegraphics[width=0.49\linewidth]{H2_CIS.pdf} \includegraphics[width=0.49\linewidth]{H2_CIS.pdf}
\hspace{0.05cm}
\includegraphics[width=0.49\linewidth]{H2_TDDFT.pdf} \includegraphics[width=0.49\linewidth]{H2_TDDFT.pdf}
\includegraphics[width=0.49\linewidth]{H2_BSE_RHF.pdf} \includegraphics[width=0.49\linewidth]{H2_BSE_RHF.pdf}
\caption{ \caption{
@ -789,6 +790,9 @@ The $D_{2h}$ and $D_{4h}$ optimized geometries of the $^1 A_g$ and $^3 A_{2g}$ s
EOM-CCSD and SF-ADC calculations have been taken from Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}. EOM-CCSD and SF-ADC calculations have been taken from Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}.
All of them have been obtained with a UHF reference like the SF-BSE calculations performed here. All of them have been obtained with a UHF reference like the SF-BSE calculations performed here.
Table~\ref{tab:CBD_D2h} shows results obtained for the $D_{2h}$ rectangular equilibrium geometry and Table~\ref{tab:CBD_D4h} shows results obtained for $D_{4h}$ square equilibrium geometry. These results are given with respect to the singlet ground state. For each geometry three states are under investigation, for the $D_{2h}$ CBD we look at the $1 ~^3 B_{1g}$, $1~^1 B_{1g}$ and $2 ~^1 A_{1g}$ states. For the $D_{4h}$ CBD we look at the $1 ~^3 A_{2g}$, $2~^1 A_{1g}$ and $1 ~^1 B_{2g}$ states. Several methods using spin-flip are compared to the spin-flip version of BSE with and without dynamical corrections. In Table~\ref{tab:CBD_D2h}, comparing the results of our work and the most accurate ADC level, i.e., SF-ADC(3) with SF-BSE@{\GOWO} we have a difference in the excitation energy of 0.017 eV for the $1 ~^3 B_{1g}$ state. This difference grows to 0.572 eV for the $1 ~^1 B_{1g}$ state and then it is 0.212 eV for the $2 ~^1 A_{1g}$ state. Adding dynamical corrections in SF-dBSE@{\GOWO} do not improve the accuracy of the excitation energies comparing to SF-ADC(3). Indeed, we have a difference of 0.052 eV for the $1 ~^3 B_{1g}$ state, 0.393 eV for the $1 ~^1 B_{1g}$ state and 0.293 eV for the $2 ~^1 A_{1g}$ state.
Now, looking at the Table~\ref{tab:CBD_D4h} and comparing SF-BSE@{\GOWO} to SF-ADC(3) we have an interesting result. Indeed, we have a wrong ordering in our first excited state, we find that the triplet state $1 ~^3 A_{2g}$ is lower in energy that the singlet state $B_{1g}$ in contrary to all of the results extracted in Refs.~\onlinecite{Manohar_2008} and Ref.~\onlinecite{Lefrancois_2015}. Then adding dynamical corrections in SF-dBSE@{\GOWO} not only improve the difference of excitation energies with SF-ADC(3) it gives the right ordering for the first excited state, meaning that we retrieve the triplet state $1 ~^3 A_{2g}$ above the singlet state $B_{1g}$. So here we have an example where the dynamical corrections are necessary to get the right chemistry.
%%% TABLE ?? %%% %%% TABLE ?? %%%
\begin{table} \begin{table}
\caption{ \caption{
@ -823,7 +827,7 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
\caption{ \caption{
Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $\text{X}\,{}^1B_{1g}$ singlet ground state. Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the $1\,{}^3A_{2g}$, $2\,{}^1A_{1g}$, and $1\,{}^1B_{2g}$ states at the $D_{4h}$ square-planar equilibrium geometry of the $\text{X}\,{}^1B_{1g}$ singlet ground state.
All the spin-flip calculations have been performed with a UHF reference. All the spin-flip calculations have been performed with a UHF reference.
\label{tab:CBD_D2h}} \label{tab:CBD_D4h}}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{lccc} \begin{tabular}{lccc}
& \mc{3}{c}{Excitation energies (eV)} \\ & \mc{3}{c}{Excitation energies (eV)} \\
@ -834,7 +838,7 @@ All of them have been obtained with a UHF reference like the SF-BSE calculations
EOM-SF-CCSD\fnm[1] & & & \\ EOM-SF-CCSD\fnm[1] & & & \\
EOM-SF-CCSD(fT)\fnm[1] & & & \\ EOM-SF-CCSD(fT)\fnm[1] & & & \\
EOM-SF-CCSD(dT)\fnm[1] & & & \\ EOM-SF-CCSD(dT)\fnm[1] & & & \\
SF-ADC(2)-s\fnm[2] & & & \\ SF-ADC(2)-s\fnm[2] & 0.265 & 1.658& 1.904 \\
SF-ADC(2)-x\fnm[2] & & & \\ SF-ADC(2)-x\fnm[2] & & & \\
SF-ADC(3)\fnm[2] & & & \\ SF-ADC(3)\fnm[2] & & & \\
SF-BSE@{\GOWO}\fnm[3] & -0.049 & 1.189 & 1.480 \\ SF-BSE@{\GOWO}\fnm[3] & -0.049 & 1.189 & 1.480 \\